Theory Of Financial Risks From Statistical Physics To Risk Management Cambridge University Press 2000.pdf

THEORY OF FINANCIAL RISKS

FROM STATISTICAL PHYSICS TO RISK

MANAGEMENT

JEAN-PHILIPPEBOUCHAUD and MARCPOTTERS

CAMBRIDGE

UNIVERSITY PRESS

THEORY OF FINANCIAL RISKS

FROM STATISTICAL PHYSICS TO RISK MANAGEMENT

+. :

This book summarizes recent theoretical developments inspired by statistical

physics in the description of the potential moves in financial markets, and its

application to derivative pricing and risk control. The possibility of accessing and

processing huge quantities of data on financial markets opens the path to new

methodologies where systematill comparison between theories and real data not

only becomes possible, but mandatory. This book takes a,physicist's point of view

of financial risk by comparing theory with experiment. Starting with important

results in probability theory the authors discuss the statistical analysis of real data,

the empiricaldetermination of statistical laws, the definition of risk, the theory of

optimal portfolio and the problem of derivatives (forward contracts, options). This

book will be of interest to physicists interested in finance, quantitative analysts in

financial institutions, risk managers and graduate students in mathematical finance.

JEAN-PHILIPPE BOUCHAUD was born in France in 1962. After studying at

the French Lyc6e in London, he graduated from the lkxAeNorrnale Supkrieure

in Paris, where he also obtained his PhD in physics. He was then appointed by the

CNRS until 1992, where he worked on diffusion in random media. After a year

spent at the Cavendish Laboratory (Cambridge), Dr Bouchaud joined the Service

de Physique de 1'Etat Condense (CEA-Saclay), where he works on the dynamics of

glassy systems and on granular media. He became interested in theoretical finance

in 1991 and founded the company Science & Finance in 1994 with J.-P. Aguilar.

His work in finance includes extreme risk control and alternative option pricing

models. He teaches statistical mechanics and finance in various Grades ~coles.

He was awarded the IBM young scientist prize in 1990 and the CNRS Silver Medal

in 1996.

Born in Belgium in 1969, MARC POTTERSholds a PhD in physics from

Princeton University and was a post-doctoral fellow at the University of Rome

La Sapienza. In 1995,he joined Science &Finance, a research company lacated in

Paris and founded by J.-P. Bouchaud and J.-P. Aguilar. Dr Potters is now Head of

Research of S&F,supervising the work of six other physics PhDs. In collaboration

with the researchers at S&F, he bas published numerous asticles in the new field

of statistical finance and worked on concrete applications of financial forecasting,

option pricing and risk control. Since 1998,he has also served as Head of Research

of Capital Fund Management, a successful fund manager applying systematic

trading strategies devised by S&F. Dr Potters teaches regularly with Dr Bouchaud

a-? ~cole

Centrale de Paris.

PUBLISHED BY THE PRESS SYNDICATE OF THE LISIVERSITY OF CAhILIRlDCE

The Pitt Building, Tmmpington Street, Cambridge,United Kingdom

CAMBRIDGE UNIVERSITY PRESS

The U~nburghBuilding, Cambridge CB2 2RU, UK

40 West 20th Street, New York, NY 10011-4211. USA

10 Stamford Road, Oakleigh, VIC 3 166. Australia

Ruiz de Alarcttn 13, 28014, Madrid, Spain

Dock House, The Watwfntnt. Cape Town 8001, South Africa

Contents

@ Jean-Philippe Bouchaud and Marc Potters 2OQO

This book is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements:

no reproduction of any part may rake place without

the written permission of Cambridge Univenity Press.

First published 2000

Reprinted 200 1

Foreword

page ix

Printed in the United Kingdom at the U~liversityPress, Cambridge

1 Probability theory: basic notions

Typeface Times llll4pt. System LKTg2, [DBD]

1.1 Introduction

1.2 Probabilities

1.2.1 Probability distributions

1.2.2 Typical values and deviations

1.2.3 Moments and characteristic function

1.2.4 Divergence of moments

A catalogue record of rhis book is available from he British Libra?

ISBN 0 521 78232 5 hardback

- asymptotic behaviour

1.3 Some useful distributions

1.3.1 Gaussian distribution

1.3.2 Log-normal distribution

1.3.3 Levy distributions and Paretian tails

1.3.4 Other distributions

1.4 Maximum of random variables

- statistics of extremes

1.5 Sums of random variables

1.5.1 ~onvoLtions

1.5.2 Additivity of cumulants and of tail amplitudes

1.5.3 Stable distributions and self-similarity

1.6 Central limit theorem

1.6.1 Convergence to a Gaussian

1.6.2 Convergence to a Uvy distribution

1.6.3 Large deviations

1.6.4 The CLT at work on a simple case

1.6.5 Truncated Evy distributions

1.6.6 Conclusion: survival and vanishing of tails

1.7 Correlations, dependence, non-stationary models

Preface

1.7.1 Correlations 36

1.7.2 Non-stationary models and dependence 36

1.8 Central liinit theorem for random matrices 39

1.9 Appendix A: non-stationarity and anomalous kurtosis 43

1.10 Appendix B: density of eigenvalues for random correlation matrices 43

1.1 1 References

3.3.1 Correlated Gaussian fluctuations

3.3.2 Tower-law' fluctuations

3.4 Optimized trading

3.5 Conclusion of the chapter

3.6 Appendix C: some useful results

3.7 References

- Hurst exponent

2.4 Anomalous kurtosis and scale fluctuations

2.5 Volatile markets and volatility markets

2.6 Statistical analysis of the forward rate curve

2.6.1 Presentation of the data and notations

2.6.2 Quantities of interest and data analysis

2.6.3 Comparison with the Vasicek model

2.6.4 Risk-prenrium and the z/B law

2.7 Correlation matrices

2.8 A simple mechanism for anomalous price statistics

2.9 A simple model with volatility correlations and tails

2.10 Conclusion

2.11 References

3 Extreme risks and optimal portfolios

3.1 Risk measurement and diversification '

3.1 .l Risk and volatility

3.1.2 Risk of loss and 'Value at Risk' (VaR)

3.1.3 Temporal aspects: drawdown and cumulated loss

4 Futures and options: fundamental concepts

4.1 Introduction

4.1.1 Aim of the chapter

4.1.2 Trading strategies and efficient markets

4.2 Futures and forwards

4.2.1 Setting the stage

4.2.2 Global financial balance

4.2.3 RisMess hedge

4.2.4 Conclusion: global balance and arbitrage

4.3 Options: definition and valuation

4.3.1 Setting the stage

4.3.2 Orders of magnitude

4.3.3 Quantitative analysis

- option price

4.3.4 Real option prices, volatility smile and 'implied' kurtosis

4.4 Optimal strategy and residual risk

4.4.1 Introduction

4.4.2 A simple case

4.4.3 General case: 'A' hedging

4.4.4 Global hedgingiinstantaneous hedging

4.4.5 Residual risk: the Black-Scholes miracle

4.4.6 Other measures of risk

- hedging and VaR

4.4.7 Hedging errors

4.4.8 Summw'

4.5 Does the price of an option depend on the mean return?

4.5.1 The ca5e of non-zero excess return

4.5.2 The Gaussian case and the Black-Scholes limit

4.5.3 Conclusion. Is the price of an option unique?

416 Conclusion of the chapter: the pitfalls of zero-risk

4.7 Appendix D: computation of the conditional mean

4.8 Appendix E: binomial model

4.9 Appendix F: option price for (suboptimal) A-hedging

4.10 References

3.1.4 Diversification and utility - satisfaction thresholds

3.1.5 Conclusion

3.2 Portfolios of uncorrelated assets

3.2.1 Uncorrelated Gaussian assets

3.2.2 Uncorrelated 'power-law' assets

3.2.3 Txponential' assets

3.2.4 General case: optimal portfolio and VaR

3.3 Portfolios of correlated assets

Statistics of real prices

2.1 Aimofthechapter

2.2 Second-order statistics

2.2.1 Variance, volatility and the additive-multiplicative crossover

2.2.2 Autocorrelation and power spectrum

2.3 Temporal evolution of fluctuations

2.3.1 Temporal evolution of probability distributions

2.3.2 Multiscaling

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